Let $F$ be a locally constant sheaf on a topological space $X$ trivialized by an open cover $\bigcup_iU_i$ with $F\vert_{U_i}\cong\Delta S_i$, where the $S_i$ are sets and $\Delta$ refers to the constant sheaf functor. Denoting $U_i\cap U_j$ by $U_{ij}$ we therefore have isomorphisms $\Delta S_i\vert_{U_{ij}}\cong F\vert_{U_{ij}}\cong\Delta S_j\vert_{U_{ij}}$. In particular for all $p\in U_{ij}$ we have an isomorphism of stalks $\left(\Delta S_i\vert_{U_{ij}}\right)_p\cong\left(\Delta S_j\vert_{U_{ij}}\right)_p$, so when $U_{ij}\neq\varnothing$ one finds an isomorphism $\theta_{ij}(p):S_i\to S_j$ since the stalks of a constant sheaf on an object are just that object.
In a section 1.1 of this paper, a covering of $X$ split by $\bigcup_iU_i$ is a locally constant sheaf trivialized by the $U_i$ such that whenever $U_{ij}\neq\varnothing$, $\theta_{ij}(p)=\theta_{ij}(q)$ for all $p,q\in U_{ij}$. In other words, the transition maps on the stalks are the same at any point.
I'm struggling to see what this condition is telling us, and why it is always true for locally connected $X$.
- What is an example of a locally constant sheaf on, say, $\mathbb{Q}$, which does not satisfy this condition?
- Why is this condition always satisfied when the space is locally connected?
I am studying this in the context of fundamental groups/groupoids/progroupoids/localic progroupoids of (Grothendieck) topoi as the paper indicates, and I'm currently trying to clarify the role of local connectedness.
Something interesting about the theory of fundamental group-like structures is that for a connected topological space (and possibly for more general connected topoi? e.g. the étale topos of a connected scheme) is that the local connectedness requirement does not appear when classifying finite covers. For example, for any connected topological space $X$, the category of finite covers is a Galois category. So I'm curious to know if maybe there is some result which says if the sheaf is locally constant then it is a cover.